Practice Test 80 - Algebra 2

20 questions. Answer them in any order, then submit to see your score and every explanation.

Let $z = 2 - 5i$ . The conjugate of $z$ is $\overline{z} = 2 + 5i$ . How are the points representing $z$ and $\overline{z}$ related geometrically on the complex plane?
A store offers a rebate that pays $160 after the first month, and each month after that the rebate payment is multiplied by $0.75$ (so it decreases). The total rebate paid over 7 months is $160+160(0.75)+160(0.75)^2+\cdots+160(0.75)^6.$ What is the total rebate over 7 months, approximately?
Consider the rational expression $R(x)=\frac{(x-1)(x+4)}{x-1}.$ If you view the numerator $(x-1)(x+4)$ as a single entity and focus on dependence, which statement is correct about where the potential restriction on $x$ comes from?
The point $P$ represents a complex number $z$ on the coordinate plane shown. If $z$ is written in polar form as $r(\cos\theta + i\sin\theta)$ where $\theta \in [0, 2\pi)$ , which expression best represents $\theta$ ?
Because polynomial identities extend to complex numbers, you can factor expressions that do not factor over the reals. Factor $x^2+1$ completely over $\mathbb{C}$ .
The concentration of a medication in the bloodstream follows $C(t) = 50e^{-0.2t}$ mg/L, where $t$ is hours after administration. A doctor needs to know when the concentration drops to 5 mg/L. After setting up $50e^{-0.2t} = 5$ , which step correctly leads to the logarithmic solution?
Factor the quadratic $x^2+25$ over the complex numbers. Recall: every quadratic $ax^2+bx+c=0$ has exactly 2 complex zeros counting multiplicity (two real, one repeated real, or a complex conjugate pair), and this idea previews the Fundamental Theorem of Algebra: a degree- $n$ polynomial has exactly $n$ complex zeros counting multiplicity. Which factorization and zero count is correct for $x^2+25$ ?
Prove the polynomial identity $(x^2 + y^2)^2 = (x^2 - y^2)^2 + (2xy)^2$ for all real $x, y$ by expanding both sides and simplifying. Which option correctly completes the proof?
For the system below (variable order $x, y, z$ ), which matrix is the coefficient matrix $A$ in $AX=B$ ? $\begin{cases} -x+4y+z=6 \\ 2x-y+3z=-1 \\ 5x+0y-2z=7 \end{cases}$
Subtract and simplify to lowest terms. This uses the same rules as subtracting fractions and shows closure under subtraction. Subtract: $\displaystyle \frac{x+1}{x^2-4}-\frac{3}{x-2}$
A radioactive sample's mass is measured every day: Day: 0, 1, 2, 3 Mass (g): 80, 72, 64.8, 58.32 From the table, what is the percent rate of decay per day?
In the term $-3xy^2$ , what is the coefficient (the numerical multiplier, including sign) of the variable part $xy^2$ ?
Solve for $t$ using logarithms: $2\cdot e^{0.4t}=25$ . Give the exact form and an approximate value (use technology).
A cosine function has been transformed such that its amplitude is 3, its period is $\pi$ , and it has been shifted up 2 units. If the function starts at its maximum value when $x = 0$ , which equation represents this function?
Write the following system as a single matrix equation $AX=B$ (use variable order $x, y$ ): $\begin{cases} 3x+2y=8\\ x-5y=-3 \end{cases}$ Which choice correctly gives $A$ , $X$ , and $B$ ?
Over $\mathbb{C}$ , the difference of squares identity can be used with $b=2i$ since $(2i)^2=-4$ . Factor $x^2+4x+8$ completely over $\mathbb{C}$ .
The function $d(t) = 3\sin(\frac{\pi t}{6}) + 5$ models the depth of water in feet at a dock, where $t$ is hours after midnight. The dock is usable when depth is at least 4 feet. Using the graph shown, what domain represents usable hours in the first day?
Show that the zeros of $r(x)=x^2+x-6$ can be found by factoring. Which choice gives the correct factorization and zeros (x-intercepts)?
For $f(x)=\frac{(x+2)(x-1)}{x^2-4},$ find the zeros, vertical asymptote(s), and any hole. (Use factorization to decide whether a common factor creates a hole.)
On the complex plane, the distance between $z=-1+2i$ and $w=3-1i$ is $|z-w|$ . What is $|z-w|$ ?

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